⠀⠀⠀⣿⣿⣿⠿⠿⠿⢿⣿⣷⢀⣴⣿⣿⠿⠿⠿⣿⣿⣦⡀⣿⣿⣿⠀⠀⠀⠀⠀⣴⣿⣿⠿⠿⠿⣿⣿⣦⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⣿⣿⠀⣿⣿⡿⠿⠿⠿⠿⠿⠀
⠀⠀⠀⣿⣿⡿⠶⠶⠶⢾⣿⣏⢸⣿⣿⡇⠀⠀⠀⢸⣿⣿⡇ ⣿⣿⣿⠀⠀⠀⠀⠀⠛⠿⠿⠿⠷⠶⢶⣶⣦⢸⣿⣿⣿⣿⣿⣿⣿⣿⡇ ⣿⣿⠀⣿⣿⡷⠶⠶⠶⠶⠀⠀
⠀⠀⠀⣿⣿⣿⣶⣶⣶⣿⣿⠿⠈⠻⢿⣿⣷⣶⣾⣿⡿⠟⠁⣿⣿⣿⣿⣿⣿⣿ ⠻⣿⣿⣶⣶⣶⣾⣿⠿⢸⣿⣿⡇⠀⠀⠀⣿⣿⣿⠀⣿⣿⠀⣿⣿⣷⣶⣶⣶⣶⣶⠀
⠀⠀⠀⠀⠀⠀⠀⠀ ⣾⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤
⠀⠀⠀⠀⠀ ⠀⠀⠠⣿⡇⠀⠀⠀⠀⡀⠀⠠⠚⠛⠻⣿⣆⠀⠀⣾⣿⠁⠀⡞⠛⠛⢷⣆⠀⠀⠀⠀⠀⣴⠃⢀⣀⣠⣤⣤⣤⣶⣶⠀⠀⢀⣤⣤⠞⠉⠁ ⢤⣶⡄
⠀⠀⠀⠀⠀⠐⠒⠚⣿⡏⢀⣴⣾⡿⠋⠀⠀⠀⠀⠀⢸⣿⠀⢠⣿⣿⠀⠈⣧⣶⣿⡟⠁⠀⠀⠀⠀⣾⡷⠟⠛⠉⠉⠁⠀⠀⢹⡟⣀⣴⣿⡿⠁⠀⠀⠀⣤⣤⣤
⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣶⣶⣶⣶⣶⡶⠦⢀⣼⡿ ⢸⣿⡇⠀⠀⣿⠛⠹⣧⠀⠀⣿⠟⠒⢲⣶⣶⠖⠒⣶⡶⢠⣿⣷⠟⠙⢿⣧⡀⠀⠀⠀⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⣀⣀⣤⡶⠾⠛⠉⠀⠀⣸⣿⡇⠀⠀⠻⠀⠀⠈⠻⣦⡄⠀⠀⠀⣾⣿⡏⠀⢠⠟⠀⣾⡟⠁⠀⠀⠀ ⠙⢿⣄⠀⠀⣿⣿⡿
⠀⠀⠀⠀⠀⠻⠟⠁⠀⠀⠀⠀⠀⠚⠋⠉⠙⢿⣷⣦⣄⣀⣤⢿⣿⣧⡀⡠⠀⠀⠀⠀⠀⠈⠻⠆⠀⠀⣿⡿⠀⢀⣠⡤⢾⡟⠀⠀⠀⠀⠀⠀⠀ ⠻⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⠀⠉⠙⠋⠁⠀⠉⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠀⠛⠁⠀⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀ ⠙⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⡶⠷⠞⠒⠛⠓⠾⠶⠿⠦⢤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⠛⠉⠁⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠈⠙⠳⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⣶⠞⠋⠀⠀⠀⢀⡤⡴⠶⠿⠛⡟⠛⠛⠻⠷⠦⠤⡖⠲⣄⠀⠀⠈⠙⠷⣖⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣰⣾⣿⠀⡘⡄⠀⠀⠁⠀⡀⠀⡴⠃⢰⡟⠀⠈⠻⣷⣤⣤⠀⠀⠉⢳⡦⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡇⠀⠀⢀⣼⣿⡟⡽⣠⣡⣃⣀⣀⣀⣀⠓⠊⣠⣔⣙⡇⠀⠀⠀⠈⢿⣿⣇⠀⠀⠀⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡇⠀⠀⢸⣿⣿⠾⡳⠋⡇⠈⠀⠙⠺⠿⠿⡿⠭⠟⠈⢹⡼⡉⠢⣀⠈⢻⡇⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣨⠟⣇⣤⠴⠋⠁⡀⠈⠀⣠⠿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⡜⠙⢌⠀⠀⠑⢄⢧⡀⣀⢸⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⢁⢔⣟⡡⠤⣲⣟⣉⣤⣼⣡⣄⣘⣆⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠳⣄⠀⠀⠀⠉⠻⣯⡉⠒⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣳⠃⢰⠈⠻⣈⣖⠲⠾⠛⠛⠉⠀⠉⢺⢟⢧⡀⠀⠀⠀⠀⣰⣃⣴⠶⠶⠶⡺⠦⣀⠀⠀⠀⠈⡎⠢⡀⠀⠐⠢⢄⡀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢱⠃⢰⠘⡀⠀⠀⢰⠀⠀⠀⠀⠈⢆⠀⢸⢸⠙⢵⡄⠀⠀⣰⣿⡟⠁⠀⠀⠀⠈⠳⣌⠙⠲⠤⣤⠇⡄⠈⣷⢦⣠⠎⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⠀⣰⢃⡾⠀⠀⢇⢱⠀⠀⢸⠈⡄⠀⠀⠀⠀⠀⠘⣾⠀⠈⠘⣇⢸⡿⡝⡇⠀⠀⠀⠀⠀⠀⠀⠑⢺⠋⡜⣰⠁⠀⢸⠎⢧⠀⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠀⢠⠃⢸⣇⠀⠀⠈⢦⣇⠀⠸⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⠀⠹⣻⢳⠃⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⣰⣱⠃⠀⠐⡞⠰⡌⣇⠀⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⢀⡏⠀⠸⣸⠀⠈⢄⠀⠻⣆⠀⠀⠀⠀⠀⠀⠀⠀⢺⢸⠀⠀⠀⣿⠈⠀⡏⡆⠀⠀⠀⠀⠀⡀⠀⢠⢧⡟⠁⢀⠂⠀⠱⡀⠳⡸⡄⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⢟⡄⠀⢎⠳⣄⢻⠀⡀⠀⠀⠀⠀⠀⠀⠀⣾⠀⠀⠀⣿⠀⠀⣿⠀⠀⠀⠀⠀⠀⠰⡀⡆⣿⣀⠔⢡⠃⠀⠀⢷⠀⢣⢳⠀⠀⠀⠀⠀⠀
⠀⠀⠀⠀⠀⢰⡃⠀⠀⠀⠸⡄⠀⠈⢢⡀⢹⢀⠃⠀⠀⠀⠀⠀⠀⢰⢻⠀⠀⠀⣿⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⢣⢡⡟⠁⡰⠃⠀⠀⡸⠘⡆⠘⣬⢧⠀⠀⠀⠀⠀
⠀⠀⠀⠀⢠⠇⠁⠀⠀⠀⠀⢡⠀⠀⠀⠱⣼⠸⠀⡀⠀⠀⠀⠀⠀⢈⢸⠀⠀⠀⣿⡼⡐⢸⠀⠀⠀⠄⠀⠀⠀⠘⡞⣧⡜⠁⠀⠀⢠⠃⠀⢧⠀⢿⠘⡆⠀⠀⠀⠀
⠀⠀⠀⠀⡜⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⣿⡇⠀⠇⠀⠀⣴⡆⠀⠀⠋⠀⠀⠀⣿⡇⠇⠋⠀⠀⡜⡶⣄⠀⠀⠀⠇⣿⡄⠀⠀⠀⠘⠀⠀⠘⠒⠋⠀⢹⡀⠀⠀⠀
⠀⠀⠀⢰⠃⠀⠀⡀⠀⠉⠲⢦⢤⣀⣀⢼⢹⠁⠀⠀⢀⣾⠟⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⡸⠁⠹⡜⣆⠀⠀⠦⣏⡗⣄⣀⡠⣴⠔⠈⠁⠀⠀⠀⠀⡇⠀⠀⠀
⠀⠀⠀⢸⠀⠀⠀⠰⡀⠀⠀⠀⠙⠢⣀⡎⡼⠀⠀⢠⣾⠋⠀⠀⠱⡄⠀⠀⠀⠀⣿⣶⠀⠀⢰⠃⠀⠀⠘⣎⢦⠀⠀⢹⠸⡄⡤⠊⠁⠀⠀⢀⡜⠀⠀⠀⣇⠀⠀⠀
⠀⠀⠀⡟⠀⠐⢤⣀⠈⠢⣀⠀⠀⠀⣸⢠⠇⠀⢠⡿⠁⠀⠀⠀⠀⠘⡄⠀⠀⠀⣿⡇⠇⢀⣎⠇⠀⠀⠀⠈⢯⢳⡀⠘⡆⢹⠀⠀⠀⢀⡴⠋⣀⡠⠄⠀⢸⠀⠀⠀
⠀⠀⠀⡇⠀⠀⠀⠈⠉⠓⠚⢿⣖⠚⡇⢸⠀⣰⡟⠁⠀⠀⠀⠀⠀⠀⠹⡀⠀⠀⢻⡷⠀⣸⢎⠀⠀⠀⠀⠀⠀⢳⡱⡄⡇⠸⡛⢲⡿⠓⠚⠉⠁⠀⠀⠀⢸⠀⠀⠀
⠀⠀⠀⡇⠀⠀⠀⠀⠈⠉⠉⣁⠔⡼⠁⢸⣶⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠀⠀⣾⣧⢠⢏⠎⠀⠀⠀⠀⠀⠀⠀⠱⡹⡇⠀⢳⡠⣌⠉⠉⠁⠀⠀⠀⠀⠸⡆⠀⠀
⠀⠀⠀⡇⠀⠀⠈⠉⠉⠉⠉⢀⠞⡇⠀⣸⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⣿⡿⣠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡄⢱⣱⠀⢸⠑⠄⠉⠉⠉⠉⠁⠀⠀⢰⠃⠀⠀
⠀⠀⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠃⠀⠀⠀⠀⠀⠀⠀⠀⢰⠁⠀⣿⠆⠈⣆⠀⠀⠀⠀⠀⠀⠀⠀ ⠈⣆⠀⠀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠰⢋⡆⠀⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠀⢀⣿⠀⠀⡏⠢⠀⠀⠀⠀⠀⠀ ⠀⠀ ⣼⠀⠀
⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⢡⣧⡐⢄⡠⣄⠀⠀⠀⣀⡀⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⢀⡠⠐⠀⡔⠀⠀⣼⣼⠀⠀⣇⠀⠀⠀⠀⠀⠀⠀⠀ ⠀⣧⠀⠀
⠀⠀⢾⠀⠀⠀⠀⠀⠀⡴⠁⠀⣸⠀⠀⣿⢻⢿⡢⢬⣓⣇⣄⣊⡀⠀⠀⠀⠀⠸⡏⣿⠀⢀⡴⠚⠁⢀⣼⣊⡠⣴⡾⡏⢸⡆⠀⣿⠀⠀⠢⡀⠀⠀ ⠀⠀⢸⠀⠀
⠀⠀⠈⡦⣢⣢⠀⣤⢊⣀⣀⠄⡿⠀⠀⣿⢸⢀⠈⠙⡟⠒⡒⠒⠛⠭⡖⠤⣧⣿⠐⡡⢒⠯⠛⠛⠛⠋⢉⡥⣾⡇⠈⡇⢹⡀⢀⣀⢈⢠⡀⣰⣔⣠⡏⠀⠀
⠀⠀⠀⠘⢷⠧⠭⣾⠋⠷⢶⠟⠀⠀⠀⢻⠻⣼⠀⠀⠁⠀⡇⠀⠀⠀⡇⠀⠀⠀⣿⣿⠀⠀⠀⡇⠀⠀⠀⠀⢸⣷⣿⠟⠀⢧⠀⠀⠹⣷⠾⠧⠽⠭⠽⣷⠏⠀⠀⠀
⠀⠀⠀⠀⠈⡇⠀⣿⣷⠀⡏⠀⠀⠀⠀⢸⠀⠈⠙⠲⠼⣶⣃⡀⠀⠀⡎⠑⠂⠀⣿⣿⠀⠀⠀⡇⢀⣀⣤⠤⠞⠋⠁⠀⠀⠸⡄⠀⠀⠘⣤⠀⡄⠀⢠⡇⠀⠀⠀⠀
⠀⠀⠀⠀⠠⣧⣀⣸⣈⣸⠁⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠈⠉⠉⠑⠒⠦⠤⠼⠯⠤⠒⠚⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⢯⣰⣳⣀⣸⠇⠀⠀⠀⠀
⠀⠀⠀⠀⠀⠀⠈⠉⠉⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠀⠀⠀⠀⠀⠀